Optimal inapproximability with universal factor graphs

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.